Stein's Example - Formal Statement

Formal Statement

The following is perhaps the simplest form of the paradox. Let θ be a vector consisting of n ≥ 3 unknown parameters. To estimate these parameters, a single measurement X_i is performed for each parameter θ_i, resulting in a vector X of length n. Suppose the measurements are independent, Gaussian random variables, with mean θ and variance 1, i.e.,

Thus, each parameter is estimated using a single noisy measurement, and each measurement is equally inaccurate.

Under such conditions, it is most intuitive (and most common) to use each measurement as an estimate of its corresponding parameter. This so-called "ordinary" decision rule can be written as

The quality of such an estimator is measured by its risk function. A commonly used risk function is the mean squared error, defined as

Surprisingly, it turns out that the "ordinary" estimator proposed above is suboptimal in terms of mean squared error. In other words, in the setting discussed here, there exist alternative estimators which always achieve lower mean squared error, no matter what the value of is.

For a given θ one could obviously define a perfect "estimator" which is always just θ, but this estimator would be bad for other values of θ. The estimators of Stein's paradox are, for a given θ, better than X for some values of X but necessarily worse for others (except perhaps for one particular θ vector, for which the new estimate is always better than X). It is only on average that they are better.

More accurately, an estimator is said to dominate another estimator if, for all values of, the risk of is lower than, or equal to, the risk of, and if the inequality is strict for some . An estimator is said to be admissible if no other estimator dominates it, otherwise it is inadmissible. Thus, Stein's example can be simply stated as follows: The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk.

Many simple, practical estimators achieve better performance than the ordinary estimator. The best-known example is the James–Stein estimator, which works by starting at X and moving towards a particular point (such as the origin) by an amount inversely proportional to the distance of X from that point.

For a sketch of the proof of this result, see Proof of Stein's example.